Since (REF ) is concave, problem (REF ) is also concave and satisfies Slater's condition [1]}. Therefore, we can solve the dual problem because the duality gap between the primal and dual problems is zero. To do so, we first provide the partial Lagrangian function of problem (REF ) with respect to () as
\(\begin{split}\mathcal {L}(\mathbf {\tau },\mathbf {\varphi },\mathbf {e},\mathbf {\lambda })& = R - \sum _{k=1}^K \lambda _k \Big ( R - R_k(\tau _k, \varphi _k, e_k) \Big )\\& = R(1 - \sum _{k=1}^K \lambda _k) + \sum _{k=1}^K \lambda _k R_k (\tau _k, \varphi _k, e_k),\end{split}\) 
